Summer School in Inverse Problems; June 2009; Newark, DE

对于数学及其应用研究所（IMA）及其参与机构（PI），首席研究员和他的同事们于6月15日至7月3日组织了IMA/PI反问题数学暑期学校。在这个项目中，调查人员包括来自参与机构以外的学生。这个项目面向数学科学的研究生。2009年夏季反问题数学课程涵盖了三个不同的主题：双曲偏微分方程的反问题、频域的反散射和反问题的变分技术。该计划涵盖了用于解决这些领域中数学研究前沿问题的技术。为期一周的讲座和解决问题的会议是由这些领域的世界领导人在非正式的环境中提出的——William Symes（波的逆问题），John Sylvester（逆散射）和Jonathan Borwein（变数分析）。三个系列讲座中的两个（逆散射和变分技术）将作为单独的章节出现在b施普林格将于2010年出版的一本书中。逆问题无处不在，从确定全球变暖的原因，到寻找地表下的石油和天然气，再到用医学成像诊断疾病。反问题的基本思想很简单：给出观察结果，确定原因。不幸的是，大多数反问题都不容易解决：观察的模型往往是不完整或不正确的，而且，即使模型与事实完美匹配，也可能无法从观察中准确地确定原因（所谓的“病态”的诅咒）。在过去的20年里，反问题的数学和解决这些问题的能力发生了巨大的变化，这主要是由计算能力的提高和通常被称为“实验数学”的同步发展引起的。在此之前，从未有过新的理论工具能够如此直接和具有实际意义地进行测试。这个IMA/PI反问题数学暑期学校旨在为未来的数学家在这个日益增长和日益重要的领域的研究做好准备。这些学校将学生与优秀的研究人员聚集在一起，在一个密集的环境中，旨在引导学生从更熟悉的以课程为导向的问题解决技能到数学研究的前沿。学生不仅可以获得解决高级和研究级课题的经验，还可以获得与不同背景的人合作的经验。2009年的项目收到了来自不同群体的51份申请，其中29份来自参与的ima机构，22份来自非ima机构。该项目支持来自非ima附属机构的学生。

LukeDMS-0852454 For the Institute for Mathematics and its Applications (IMA)and its Participating Institutions (PI), the principalinvestigator and his colleagues organize the IMA/PI Summer Schoolon the Mathematics of Inverse Problems, June 15-July 3. In thisproject the investigators include students who are not from theparticipating institutions. The program is geared towardgraduate students in the mathematical sciences. The 2009 summerprogram on the mathematics of inverse problems covers threedifferent themes: inverse problems for hyperbolic partialdifferential equations, inverse scattering in the frequencydomain, and variational techniques for inverse problems. Theprogram covers the techniques used to tackle problems at thecutting edge of mathematical research in each of these areas. Week-long lectures and problem-solving sessions are presented inan informal environment by world leaders in each of these areas-- William Symes (inverse problems for waves), John Sylvester(inverse scattering), and Jonathan Borwein (variationalanalysis). Two of the three lecture series (inverse scatteringand variational techniques) are under contract to appear asseparate chapters in a book to be published by Springer in 2010. Inverse problems are everywhere, from determining the causesof global warming, to finding oil and natural gas under theearth's surface, to diagnosing diseases with medical imaging. The basic idea of an inverse problem is simple: given anobservation, determine the cause. Unfortunately, most inverseproblems are not easily solved: the model for the observation isoften incomplete or incorrect, and, even if the model is aperfect match to the truth, it may be impossible to accuratelydetermine the cause from an observation (the curse of so-called"ill-posedness"). The last 20 years has seen a dramatic shift inthe mathematics of inverse problems and the capabilities forsolving them, initiated largely by improvements in computingpower and the concurrent evolution of what is often called"experimental mathematics." Never before have new theoreticaltools been able to be tested with such immediacy and practicalimport. This IMA/PI Summer School on the Mathematics of InverseProblems aims at preparing future mathematicians for research inthis growing and increasingly vital field. The schools bringstudents together with outstanding researchers in an intensivesetting that is intended to lead students from the more familiarcourse-oriented problem solving skills to the frontiers ofmathematical research. Students gain experience not only inattacking problems on advanced and research-grade topics, butalso in working collaboratively with people from differentbackgrounds. The 2009 program has received 51 applications froma diverse group of individuals, 29 from participating IMAinstitutions and 22 from non-IMA institutions. This projectsupports students from non-IMA affiliated institutions.